Does Judaism believe in Reincarnation?

Reincarnation (gilgul) has become a common belief in today's orthodox Judaism. However, this was not always so. Daf Yomi just finished the last chapter of Sanhedrin which deals with the principles of faith including, Messiah, World to come, reward and punishment. One thing noticeably missing is reincarnation.  In fact, reincarnation is not mentioned anywhere in the Talmud, or Midrashim. R' Saadya Gaon said the following about reincarnation:

Yet, I must say that I have found certain people, who call themselves Jews, professing the doctrine of metempsychosis (reincarnation) which is designated by them as the theory of “transmigration” of souls. What the mean thereby is that the spirit of Ruben is transferred to Simon and afterwards to Levi and after that to Judah. Many of them would go so far as to assert that the spirit of a human being might enter into the body of a beast or that of a beast into the body of a human being, and other such nonsense and stupidities.

"Who call themselves Jews", "Nonsense and stupidities" this goes beyond just saying he had a lack of tradition and "was not convinced". Rav Saadia Gaon denies it even deserves a place in a discussion of Judaism:

I was wary about my words to mention their opinion, and they are worthy of such [neglect] because of their severe flaws, if I were not afraid of the foolish.

The Gemara in Bava Metziah (107a) comments on the pasuk ברוך אתה בבואך ברוך אתה בצאתך that it is teaching us that just like a person enters the world without sin he should leave the world without sin. The רש"ש there comments that this contradicts gilgulim. The reason being, that the premise of a gilgul is that a nefesh that already sinned comes into the world to be מתקן that sin. However, the gemara states explicitly that a person comes into the world without sin. 

So why does everyone now believe in reincarnation. The answer is that reincarnation is an important concept in Kabbalah. The Zohar and related literature are filled with references to reincarnation, addressing such questions as which body is resurrected and what happens to those bodies that did not achieve final perfection, how many chances a soul is given to achieve completion through reincarnation, whether a husband and wife can reincarnate together, if a delay in burial can affect reincarnation, and if a soul can reincarnate into an animal. The acceptance and proliferation of Kabbalah has cause spread this concept and caused it to be widely accepted.

This dispute raises a number of very difficult questions:

1. How is it that such a fundamental concept like reincarnation is not mentioned anywhere in the Talmud?
2. How can there be a dispute about such a fundamental point, whether souls reincarnate?
3. How could such a great Gaon like R' Saadya Gaon not know about reincarmnation and deny it's existence?

We can only conclude that reincarnation, like many other ideas, has no ancient mesora and in fact Reincarnation is a foreign idea, imported into Judaism from other religions, such as Hinduism and Buddhism, where reincarnation really is a major tenet of the faith.